reserve X for set;

theorem Th6:
  for S being SigmaField of X, M being sigma_Measure of S, G,F
being sequence of S st (M.(F.0) <+infty & G.0 = {} & for n being Nat
 holds G.(n+1) = F.0 \ F.n & F.(n+1) c= F.n ) holds M.(meet rng F) = M.(F.0)
  - M.(union rng G)
proof
  let S be SigmaField of X, M be sigma_Measure of S, G,F be sequence of S;
  assume that
A1: M.(F.0) <+infty and
A2: G.0 = {} & for n being Nat holds G.(n+1) = F.0 \ F.n & F.
  (n+1) c= F .n;
A3: union rng G = F.0 \ meet rng F by A2,Th4;
A4: M.(F.0 \ union rng G) = M.(meet rng F) by A2,Th5;
  M.(F.0 \ meet rng F) <> +infty by A1,MEASURE1:31,XBOOLE_1:36;
  then M.(union rng G) <+infty by A3,XXREAL_0:4;
  hence thesis by A3,A4,MEASURE1:32,XBOOLE_1:36;
end;
